Answer :
Let's solve the problem step by step to find the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] given the function [tex]\(f(x) = 4|x - 5| + 3\)[/tex].
1. Set the function equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value:
Subtract 3 from both sides:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x - 5| = 3
\][/tex]
4. Consider the two cases for the absolute value equation:
Case 1: [tex]\(x - 5 = 3\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
Case 2: [tex]\(x - 5 = -3\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
5. Conclusion:
The values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] are [tex]\(x = 8\)[/tex] and [tex]\(x = 2\)[/tex].
Therefore, the correct answer is [tex]\(x = 2\)[/tex] and [tex]\(x = 8\)[/tex].
1. Set the function equal to 15:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value:
Subtract 3 from both sides:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x - 5| = 3
\][/tex]
4. Consider the two cases for the absolute value equation:
Case 1: [tex]\(x - 5 = 3\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
Case 2: [tex]\(x - 5 = -3\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
5. Conclusion:
The values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 15\)[/tex] are [tex]\(x = 8\)[/tex] and [tex]\(x = 2\)[/tex].
Therefore, the correct answer is [tex]\(x = 2\)[/tex] and [tex]\(x = 8\)[/tex].
